Computer algebra systems are now ubiquitous in all areas of science and engineering. This highly successful textbook, widely regarded as the “bible of computer algebra”, gives a thorough introduction to the algorithmic basis of the mathematical engine in computer algebra systems. Designed to accompany oneor two-semester courses for advanced undergraduate or graduate students in computer science or mathematics, its comprehensiveness and reliability has also made it an essential reference for professionals in the area. Special features include: detailed study of algorithms including time analysis; implementation reports on several topics; complete proofs of the mathematical underpinnings; and a wide variety of applications (among others, in chemistry, coding theory, cryptography, computational logic, and the design of calendars and musical scales). A great deal of historical information and illustration enlivens the text. In this third edition, errors have been corrected and much of the Fast Euclidean Algorithm chapter has been renovated.

Modern computer algebra

In a data-driven society, individuals and companies encounter numerous situations where private information is an important resource. How can parties handle confidential data if they do not trust everyone involved? This text is the first to present a comprehensive treatment of unconditionally secure techniques for multiparty computation (MPC) and secret sharing. In a secure MPC, each party possesses some private data, while secret sharing provides a way for one party to spread information on a secret such that all parties together hold full information, yet no single party has all the information. The authors present basic feasibility results from the last 30 years, generalizations to arbitrary access structures using linear secret sharing, some recent techniques for efficiency improvements, and a general treatment of the theory of secret sharing, focusing on asymptotic results with interesting applications related to MPC.

Secure Multiparty Computation and Secret Sharing

In this work, we propose two McEliece variants: one from Moderate Density Parity-Check (MDPC) codes and another from quasi-cyclic MDPC codes. MDPC codes are LDPC codes of higher density (and worse error-correction capability) than what is usually adopted for telecommunication applications. However, in cryptography we are not necessarily interested in correcting many errors, but only a number which ensures an adequate security level. By this approach, we reduce under certain hypotheses the security of the scheme to the well studied decoding problem. Furthermore, the quasi-cyclic variant provides extremely compact-keys (for 80-bits of security, public-keys have only 4801 bits).

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MDPC-McEliece: New McEliece variants from Moderate Density Parity-Check codes

Cryptosystems I.- On Ideal Lattices and Learning with Errors over Rings.- Fully Homomorphic Encryption over the Integers.- Converting Pairing-Based Cryptosystems from Composite-Order Groups to Prime-Order Groups.- Fully Secure Functional Encryption: Attribute-Based Encryption and (Hierarchical) Inner Product Encryption.- Obfuscation and Side Channel Security.- Secure Obfuscation for Encrypted Signatures.- Public-Key Encryption in the Bounded-Retrieval Model.- Protecting Circuits from Leakage: the Computationally-Bounded and Noisy Cases.- 2-Party Protocols.- Partial Fairness in Secure Two-Party Computation.- Secure Message Transmission with Small Public Discussion.- On the Impossibility of Three-Move Blind Signature Schemes.- Efficient Device-Independent Quantum Key Distribution.- Cryptanalysis.- New Generic Algorithms for Hard Knapsacks.- Lattice Enumeration Using Extreme Pruning.- Algebraic Cryptanalysis of McEliece Variants with Compact Keys.- Key Recovery Attacks of Practical Complexity on AES-256 Variants with up to 10 Rounds.- IACR Distinguished Lecture.- Cryptography between Wonderland and Underland.- Automated Tools and Formal Methods.- Automatic Search for Related-Key Differential Characteristics in Byte-Oriented Block Ciphers: Application to AES, Camellia, Khazad and Others.- Plaintext-Dependent Decryption: A Formal Security Treatment of SSH-CTR.- Computational Soundness, Co-induction, and Encryption Cycles.- Models and Proofs.- Encryption Schemes Secure against Chosen-Ciphertext Selective Opening Attacks.- Cryptographic Agility and Its Relation to Circular Encryption.- Bounded Key-Dependent Message Security.- Multiparty Protocols.- Perfectly Secure Multiparty Computation and the Computational Overhead of Cryptography.- Adaptively Secure Broadcast.- Universally Composable Quantum Multi-party Computation.- Cryptosystems II.- A Simple BGN-Type Cryptosystem from LWE.- Bonsai Trees, or How to Delegate a Lattice Basis.- Efficient Lattice (H)IBE in the Standard Model.- Hash and MAC.- Multi-property-preserving Domain Extension Using Polynomial-Based Modes of Operation.- Stam's Collision Resistance Conjecture.- Universal One-Way Hash Functions via Inaccessible Entropy.- Foundational Primitives.- Constant-Round Non-malleable Commitments from Sub-exponential One-Way Functions.- Constructing Verifiable Random Functions with Large Input Spaces.- Adaptive Trapdoor Functions and Chosen-Ciphertext Security.

Advances in cryptology -- EUROCRYPT 2010 : 29th Annual International Conference on the Theory and Applications of Cryptographic Techniques, French Riviera, May 30-June 3, 2010 : proceedings

Introduction to coding theory (2nd edition), by J.H. van Lint. Pp.183 DM86. 1992. ISBN 0-387-54894-7 (Springer)

The McEliece cryptosystem is one of the oldest public-key cryptosystems ever designed. It is also the first public-key cryptosystem based on linear error-correcting codes. Its main advantage is to have very fast encryption and decryption functions. However it suffers from a major drawback. It requires a very large public key which makes it very difficult to use in many practical situations. A possible solution is to advantageously use quasi-cyclic codes because of their compact representation. On the other hand, for a fixed level of security, the use of optimal codes like Maximum Distance Separable ones allows to use smaller codes. The almost only known family of MDS codes with an efficient decoding algorithm is the class of Generalized Reed-Solomon (GRS) codes. However, it is well-known that GRS codes and quasi-cyclic codes do not represent secure solutions. In this paper we propose a new general method to reduce the public key size by constructing quasi-cyclic Alternant codes over a relatively small field like ${\mathbb{F}}_{2^8}$. We introduce a new method of hiding the structure of a quasi-cyclic GRS code. The idea is to start from a Reed-Solomon code in quasi-cyclic form defined over a large field. We then apply three transformations that preserve the quasi-cyclic feature. First, we randomly block shorten the RS code. Next, we transform it to get a Generalised Reed Solomon, and lastly we take the subfield subcode over a smaller field. We show that all existing structural attacks are infeasible. We also introduce a new NP-complete decision problem called quasi-cyclic syndrome decoding. This result suggests that decoding attack against our variant has little chance to be better than the general one against the classical McEliece cryptosystem. We propose a system with several sizes of parameters from 6,800 to 20,000 bits with a security ranging from 280 to 2120.

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Reducing Key Length of the McEliece Cryptosystem

In this paper we propose a new approach to investigate the security of the McEliece cryptosystem. We recall that this cryptosystem relies on the use of error-correcting codes. Since its invention thirty years ago, no efficient attack had been devised that managed to recover the private key. We prove that the private key of the cryptosystem satisfies a system of bi-homogeneous polynomial equations. This property is due to the particular class of codes considered which are alternant codes. We have used these highly structured algebraic equations to mount an efficient key-recovery attack against two recent variants of the McEliece cryptosystems that aim at reducing public key sizes. These two compact variants of McEliece managed to propose keys with less than 20,000 bits. To do so, they proposed to use quasi-cyclic or dyadic structures. An implementation of our algebraic attack in the computer algebra system Magma allows to find the secret-key in a negligible time (less than one second) for almost all the proposed challenges. For instance, a private key designed for a 256-bit security has been found in 0.06 seconds with about 217.8 operations.

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Algebraic cryptanalysis of mceliece variants with compact keys

/pdf/a-distinguisher-for-high-rate-mceliece-cryptosystems-1djgwvwlcj.pdf

A Distinguisher for High Rate McEliece Cryptosystems.

We cryptanalyse here two variants of the McEliece cryptosystem based on quasi-cyclic codes. Both aim at reducing the key size by restricting the public and secret generator matrices to be in quasi-cyclic form. The first variant considers subcodes of a primitive BCH code. The aforementioned constraint on the public and secret keys implies to choose very structured permutations. We prove that this variant is not secure by producing many linear equations that the entries of the secret permutation matrix have to satisfy by using the fact that the secret code is a subcode of a known BCH code. This attack has been implemented and in all experiments we have performed the solution space of the linear system was of dimension one and revealed the permutation matrix. The other variant uses quasi-cyclic low density parity-check (LDPC) codes. This scheme was devised to be immune against general attacks working for McEliece type cryptosystems based on LDPC codes by choosing in the McEliece scheme more general one-to-one mappings than permutation matrices. We suggest here a structural attack exploiting the quasi-cyclic structure of the code and a certain weakness in the choice of the linear transformations that hide the generator matrix of the code. This cryptanalysis adopts a polynomial-oriented approach and basically consists in searching for two polynomials of low weight such that their product is a public polynomial. Our analysis shows that with high probability a parity-check matrix of a punctured version of the secret code can be recovered with time complexity O(n
 3) where n is the length of the considered code. The complete reconstruction of the secret parity-check matrix of the quasi-cyclic LDPC codes requires the search of codewords of low weight which can be done with about 237 operations for the specific parameters proposed.

Cryptanalysis of Two McEliece Cryptosystems Based on Quasi-Cyclic Codes

Polar codes form a very powerful family of codes with a low complexity decoding algorithm that attains many information theoretic limits in error correction and source coding. These codes are closely related to Reed-Muller codes because both can be described with the same algebraic formalism, namely they are generated by evaluations of monomials. However, finding the right set of generating monomials for a polar code which optimises the decoding performances is a nontrivial task and is channel dependent. The purpose of this paper is to reveal some universal properties of these monomials. We will namely prove that there is a way to define a nontrivial (partial) order on monomials so that the monomials generating a polar code devised for a binary-input symmetric channel always form a decreasing set. We call such codes decreasing monomial codes. The fact that polar codes are decreasing monomial codes turns out to have rather deep consequences on their structure. Indeed, we show that decreasing monomial codes have a very large permutation group by proving that it contains a group called lower triangular affine group. Furthermore, the codewords of minimum weight correspond exactly to the orbits of the minimum weight codewords that are obtained from evaluations of monomials of the generating set. In particular, it gives an efficient way of counting the number of minimum weight codewords of a decreasing monomial code and henceforth of a polar code.

https://hal.inria.fr/hal-01410210/document

Ayoub Otmani

Papers

Reducing Key Length of the McEliece Cryptosystem

Algebraic cryptanalysis of mceliece variants with compact keys

A Distinguisher for High Rate McEliece Cryptosystems.

Cryptanalysis of Two McEliece Cryptosystems Based on Quasi-Cyclic Codes

Algebraic properties of polar codes from a new polynomial formalism